Meditations on a Nautical Problem, Which the Most Illustrious King Proposed to the Parisian Academy of Science
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Euler's 1728 Essay to the French Royal Academy : E004 Meditationes super problem nautico,....... Translated & Annotated by Ian Bruce 1. MEDITATIONS ON A NAUTICAL PROBLEM, WHICH THE MOST ILLUSTRIOUS KING PROPOSED TO THE PARISIAN ACADEMY OF SCIENCE Indeed in everything we draw upon, we are led by the desire of acquiring knowledge and skill, and in which we consider beautiful to excell. M. T. Cicero de Officiis THE PROBEM : WHAT IS THE BEST WAY TO PLACE THE MASTS ON VESSELS, AS FAR AS THE RELATION BETWEEN THE POSITION AS WELL AS THE NUMBER AND HEIGHT OF THE MASTS IS CONCERNED. 1. Sailing generally depends on being driven forwards chiefly by sails alone, and not by oars, arising from the arrangement and placing together of the masts. Sails usually are connected to the masts by yardarms, and on being turned into the wind they drive the ship forwards on account of that force sustained. The implantation of the masts in the ship is to be investigated , so that a ship in general can proceed with the maximum speed, and in order that this can happen, the position, height, and number of masts is to be attended to in the most diligent manner. Because in the first place in this determination it is held that much study and labour be called upon so that, if the use of the rudder cannot be avoided, then the force of that must impede the motion of the ship as little as possible, as the action of the rudder always detracts from the speed of the ship. A line drawn in ships above the bilge water from the prow to the stern is called the keel of the ship, and in French, la quille, and into this the masts are inserted as in some manner to be in the middle of the ship. If the ship is moving along the direction of its keel, there is no need for the use of the rudder for the ship to be continuing in this situation, however the masts have been implanted in the keel. Now when the ship is not moving on a par with the keel, but the direction of motion of the ship makes an angle with the keel, and which angle of deviation is called in French, l'angle de la derive, [the leeway angle] then thus, that ship does not maintain the same angle of deviation or the same position, in whatever manner the masts are situated, but in order that the angle can be retained a special placing of the masts can be determined, which must then be the single position of the masts for any other angle of deviation. And thus with ships moving forwards in water, so that they arrive at the place of choice, now they are obliged to take one deviation and then another, to which some other position of the masts would have to be attributed. But since in ships with masts once put in place, as that cannot happen with the masts remaining immobile, this change is brought about with the aid of the rudder, so that the ship is maintained at the same angle of deviation. 2. But since the rudder must act, by which the resistance to the ship has to increase, and thus the speed of the ship is diminished, and there that is affected more and more by the rudder, therefore as it were by how much the position of the masts differs more from that position in which there is no need for the rudder. Hence lest the force of the rudder should increase excessively in the deviations of the ship that occur most often, such a position must be assigned to the masts in which the rudder is said not to be in use, so that Euler's 1728 Essay to the French Royal Academy : E004 Meditationes super problem nautico,....... Translated & Annotated by Ian Bruce 2. it is apparent that the speed does not decrease sensibly at any time arising from the action of the rudder. 3. Now however many masts are put in the ship, there is always a point in the keel of the ship where if the a mast is arranged with a single height which is equal to the sum of the heights of all the other masts and equiped with just as many sails, which produces the same effect, that point can be called the common centre of the forces propelling the ship. Now from the given position of the masts and from the forces exchanged from the wind with the aid of the sails, this centre is found easily, not in an unlike manner from that , in which the common centre of gravity of bodies placed on the same line is found, yet with this difference, because here the capacity of the sails of the masts is taken in place in which the weight of the body is taken in determining the centre of gravity of bodies ; and thus it is easier from the common centre of forces moving the ship to find position of the masts : and thus it what follows it suffices to determine a single centre, for this I observe, whatever number of masts should be inserted, they can be represented readily in the same way. 4. More masts are not inserted into ships, unless the height required is of so great a size that it cannot be had from a single mast, for then it is to be effected by more than should be made available from a single mast : hence since the height of the masts is sought, the height is not to be determined unless with a single mast, but rather by several equivalent masts acting together. For this height, when it becomes known, is to be distributed over all the parts, then those parts become so small or of such heights as masts of that kind can have ; and thus the number of masts may be found, and the position of these by the previous paragraph. [This argument about the resistance of the rudder originated first from Johan Bernoulli and has been described in the book : Essay d'Une nouvelle theorie de la manoeuvre des vaisseaux. (Basle 1714). This work forms the first part of Book II of his Opera Omnia] 5. Now the height of the masts is to be determined by the capacity of the sails, which are the particular cause of the impelling force. Therefore a question has to be considered not only regarding the height of the masts, but concerning the height of the sails as well : indeed the height of the sails need not be contemplated, if only the force moving the ship along is considered, and indeed with the same force of propulsion remaining, we can consider whether that should be applied either altogether at a single point, or at several points separately, or in places with higher or lower masts. Now there a part of the force of the wind which inclines the ship forwards and immerses the prow deeper, which with higher masts in place increases the force that is applied to that effect: hence it is better with the sails made wider, in order that a sufficient amount of the force can be taken by the lower sails in place. Indeed if the higher sails are made shorter and with smaller widths then they extend themselves, and thus the force inclining the ship forwards increases; now since that is the case, the avoidance of this must be a proposition in the determination of the heights of the masts. Concerning how many and how high the masts should be made, there is a restriction put on this : the greatest attribute is the width, as long as the lower sails can be attached to the masts in place, unless here the force of the Euler's 1728 Essay to the French Royal Academy : E004 Meditationes super problem nautico,....... Translated & Annotated by Ian Bruce 3. wind is sensibly diminished, and some number of other circumstances regarding the sails allow that to happen. 6. Now this is to be observed lest the number of the sails be increased as you wish, for with too great an increase in the number of sails, it can happen that even if the ship is not capsized in the water, yet the prow is immersed further than the safety of the ship permits. In order that this is better understood, it is to be noted, whatever power of the wind applied to the sails, the force exerts a twofold effect on the ship, the one which propels the ship, and the other which inclines the ship, with the prow being immersed deeper ; one knows that this happens, as from the ship at rest being vertical, now while in motion it is inclined towards the prow, and thus with that the more with the greater wind, and in which the sails are placed at a higher point on the masts; from which it can come about that the propelling force of the wind be applied either higher or lower, so that the prow can be immersed further than is safe, or completely submerged. 7. Therefore lest the ship be excessively inclined, a limit has to be put in place how far the prow can be immersed without endangering the ship, and with which known, it is to be asked how great a force can be taken from the wind so that the ship can be inclined as far as this and no further, from which the greatest force is found by which the ship can be moved forwards, for is a greater is assumed, the ship might be at risk, because then is inclined further than is suitable: but if now the force is taken smaller, then the ship is able to move forwards at this stage without danger; hence in this way the maximum force propelling the ship is found, or the method of setting the masts in position is found, so that the ship can proceed with the greatest speed possible.